Finite Group Databasebeta

← #35224985

Subgroups — #35224985

Order 20864.

656 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 141 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104762C212yesmaximal, complemented
2C2×#19104759C212yesmaximal, complemented
3#23782214C211yesmaximal, characteristic, complemented
4#127×D163C211yesmaximal, characteristic, complemented
5#23782055C211yesmaximal, characteristic, complemented
6#23782054C211yesmaximal, characteristic, complemented
7#23782052C211yesmaximal, characteristic, complemented
8#23782040C212yesmaximal, complemented
9#23782036C212yesmaximal, complemented
10#23781617C211yesmaximal, characteristic, complemented
11#496×C163C211yesmaximal, characteristic, complemented
12C2×#14579700C2211yescharacteristic
13C2×#14579698C2211yescharacteristic
14C2×D4×D163—22no—
15C2×D4×D163C2212yes—
16#19104762—28no—
17#19104759—28no—
18C2×#14579688C2212yescomplemented
19C2×#14579687C2212yescomplemented
20C2×#14579686C2212yescomplemented
21C2×D1304C2212yescomplemented
22C2×#14579684C2212yescomplemented
23C2×#14579684C2212yescomplemented
24C2×#14579682C2212yes—
25#19104733C2211yescharacteristic, complemented
26#19104732C2211yescharacteristic
27#19104731—22no—
28#19104731C2212yes—
29#19104731C2211yescharacteristic, complemented
30#19104723C2211yescharacteristic, complemented
31#19104708—22no—
32(C22⋊D163)⋊C4—22no—
33C42×D163C2211yescharacteristic
34#19104646C2211yescharacteristic, complemented
35C8×Dic163C2212yescomplemented
36C2×SD16×C163C2212yescomplemented
37C2×D8×C163C2212yescomplemented
38#127×C163C2211yescharacteristic
39#124×C163C2211yescharacteristic, complemented
40#97×C163C2211yescharacteristic, complemented
41C23×D163—22no—
42#14579700—24no—
43#14579698—24no—
44C2×Dic326C2311yescharacteristic
45C2×D652—21no—
46C2×D652—22no—
47C2×D652C2312yes—
48C2×D652C2311yescharacteristic
49C2×C22⋊D163—21no—
50C2×C22⋊D163—24no—
51C2×C22⋊D163—22no—
52C2×C22⋊D163—22no—
53D4×D163—24no—
54D4×D163—24no—
55D4×D163—48no—
56C22×Dic163—21no—
57C2×C4×D163—21no—
58C2×C4×D163D412yes—
59C2×C4×D163C2311yescharacteristic
60#14579688—24no—
61#14579687—24no—
62#14579686—24no—
63D1304—24no—
64#14579684—24no—
65#14579684—24no—
66#14579682D418yescomplemented
67C2×C163⋊C8C2312yes—
68Dic163⋊C4—22no—
69(C2×D163)⋊C4—22no—
70C4×Dic163D412yes—
71C4×Dic163C2311yescharacteristic
72C2×Q8×C163C2311yescharacteristic
73C2×D4×C163—22no—
74C2×D4×C163C2312yes—
75C2×D4×C163C2311yescharacteristic
76SD16×C163—24no—
77D8×C163—24no—
78C2×C8×C163C2312yes—
79C22⋊C4×C163—22no—
80C42×C163C2311yescharacteristic
81C22×D163—21no—
82C22×D163—22no—
83C22×D163—48no—
84C22×D163—44no—
85C22×D163C2×D411yescharacteristic
86Dic326—22no—
87D652—22no—
88D652—22no—
89D652—22no—
90D652—44no—
91D652—42no—
92C22⋊D163—42no—
93C22⋊D163—44no—
94C22⋊D163—48no—
95C22⋊D163—44no—
96C2×Dic163—21no—
97C2×Dic163—42no—
98C2×Dic163C2×D411yescharacteristic
99C2×Dic163C2×D412yes—
100C4×D163—24no—
101C4×D163—42no—
102C4×D163C2×D414yes—
103C163⋊C8C2×D414yescomplemented
104C23×C163—21no—
105C23×C163—22no—
106Q8×C163—22no—
107D4×C163—22no—
108D4×C163—22no—
109D4×C163—22no—
110D4×C163—44no—
111C2×C4×C163—21no—
112C2×C4×C163C2×D412yes—
113C2×C4×C163C2411yescharacteristic
114C8×C163C2×D414yescomplemented
115C22×C163—41no—
116C22×C163—42no—
117C22×C163—44no—
118C22×C163—42no—
119C22×C163C22×D411yescharacteristic
120C2×D163—22no—
121C2×D163—41no—
122C2×D163—42no—
123C2×D163—44no—
124C2×D163—42no—
125C2×D163—88no—
126C2×D163#12712yes—
127C4×C163—22no—
128C4×C163—41no—
129C4×C163C22×D412yes—
130Dic163—41no—
131Dic163#12712yes—
132Dic163AGL(1,ℤ/8)14yescomplemented
133C2×C163—82no—
134C2×C163—81no—
135C2×C163C2×#12711yescharacteristic
136C2×C163C2×AGL(1,ℤ/8)12yes—
137D163—22no—
138D163—81no—
139D163—82no—
140C163#315111yescharacteristic, complemented, Sylow
141#3151—1631nomaximal, Sylow
142C2×AGL(1,ℤ/8)—1632no—
143C2×AGL(1,ℤ/8)—1632no—
144#532—1631no—
145C2×#127—1631no—
146#496—1631no—
147#496—1631no—
148#496—1631no—
149#496D16311yescharacteristic, complemented
150#493—1632no—
151#492—1632no—
152C4×M16—1631no—
153C2×#41—1631no—
154C2×#41—1631no—
155C22×D4—3262no—
156C22×D4—1632no—
157AGL(1,ℤ/8)—3268no—
158AGL(1,ℤ/8)—3268no—
159C2×SD16—1632no—
160C2×SD16—1632no—
161C2×SD16—1632no—
162C2×SD16C2×D16312yescomplemented
163C2×D8—1632no—
164C2×D8—1632no—
165C2×D8—1632no—
166C2×D8C2×D16312yescomplemented
167C2×M16—1632no—
168C4⋊Q8—1631no—
169#127—3262no—
170#127—1632no—
171#127—1631no—
172#127—1631no—
173#127C2×D16311yescharacteristic
174#124—1631no—
175#124C2×D16311yescharacteristic, complemented
176#121—3262no—
177C4×D4—3262no—
178C2×C42—1631no—
179#97—1631no—
180#97C2×D16311yescharacteristic, complemented
181C4×C8—1632no—
182C24—3262no—
183#41—3264no—
184#41—3264no—
185C2×Q8—1631no—
186C2×Q8C22×D16311yescharacteristic
187C2×D4—22no—
188C2×D4—3261no—
189C2×D4—3261no—
190C2×D4—3264no—
191C2×D4—3264no—
192C2×D4—3262no—
193C2×D4—3262no—
194C2×D4—3264no—
195C2×D4—3262no—
196C2×D4—6528no—
197C2×D4—1631no—
198C2×D4—1632no—
199C2×D4C22×D16311yescharacteristic
200C2×D4C22×D16312yes—
201C22×C4—3261no—
202C22×C4—3261no—
203C22×C4—1632no—
204C22×C4—1631no—
205SD16—24no—
206SD16—3264no—
207SD16—3264no—
208SD16—3264no—
209D8—24no—
210D8—3264no—
211D8—3264no—
212D8—3264no—
213M16—1638no—
214C2×C8—1632no—
215C2×C8C22×D16312yes—
216C4⋊C4—3262no—
217C22⋊C4—22no—
218C22⋊C4—3262no—
219C42—1632no—
220C42—1631no—
221C42C22×D16311yescharacteristic
222C23—21no—
223C23—22no—
224C23—3261no—
225C23—3262no—
226C23—6528no—
227C23—6524no—
228C23—1631no—
229Q8—22no—
230Q8—3262no—
231D4—22no—
232D4—22no—
233D4—22no—
234D4—44no—
235D4—3262no—
236D4—3262no—
237D4—3262no—
238D4—6524no—
239D4—6522no—
240D4—6522no—
241D4—6524no—
242D4—6528no—
243D4—6524no—
244C2×C4—21no—
245C2×C4—3264no—
246C2×C4—3261no—
247C2×C4—6522no—
248C2×C4—6522no—
249C2×C4—1632no—
250C2×C4—1631no—
251C2×C4—1634no—
252C2×C4D4×D16312yes—
253C2×C4C23×D16311yescharacteristic
254C8—1634no—
255C8D4×D16314yescomplemented
256C22—44no—
257C22—42no—
258C22—41no—
259C22—42no—
260C22—13048no—
261C22—3262no—
262C22—6524no—
263C22—6522no—
264C22—6521no—
265C22—6522no—
266C22—1632no—
267C22C2×D4×D16311yescharacteristic
268C4—22no—
269C4—41no—
270C4—6521no—
271C4—1634no—
272C4—1632no—
273C4C2×D4×D16312yes—
274C2—82no—
275C2—81no—
276C2—13042no—
277C2—13041no—
278C2—3262no—
279C2#127×D16311yescharacteristic
280C2AGL(1,ℤ/8)×D16312yes—
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#496×C163C2
socleC22×C163C22×D4
hypercenter#496D163
hypocenterC163#3151
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#19104762C2semidirect
2. C2×#19104759C2semidirect
3. #23782214C2semidirect
4. #127×D163C2semidirect
5. #23782055C2semidirect
6. #23782054C2semidirect
7. #23782052C2semidirect
8. #23782040C2semidirect
9. #23782036C2semidirect
10. #23781617C2semidirect
11. #496×C163C2semidirect
18. C2×#14579688C22semidirect
19. C2×#14579687C22semidirect
20. C2×#14579686C22semidirect
21. C2×D1304C22semidirect
22. C2×#14579684C22semidirect
23. C2×#14579684C22semidirect
25. #19104733C22semidirect
29. #19104731C22semidirect
30. #19104723C22semidirect
34. #19104646C22semidirect
35. C8×Dic163C22semidirect
36. C2×SD16×C163C22semidirect
37. C2×D8×C163C22semidirect
39. #124×C163C22semidirect
40. #97×C163C22semidirect
66. #14579682D4semidirect
103. C163⋊C8C2×D4semidirect
114. C8×C163C2×D4semidirect
132. Dic163AGL(1,ℤ/8)semidirect
140. C163#3151semidirect
149. #496D163semidirect
162. C2×SD16C2×D163semidirect
166. C2×D8C2×D163semidirect
175. #124C2×D163semidirect
180. #97C2×D163semidirect
255. C8D4×D163semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#191047622
C2×#191047592
#237822142
#127×D1632
#237820552
#237820542
#237820522
#237820402
#237820362
#237816172
#496×C1632
#3151163